Computer scientists have long known that randomness can be used to improve
the performance of algorithms. A familiar application is the process of
dimension reduction, in which a random map transports data from a
high-dimensional space to a lower-dimensional space while approximately
preserving some geometric properties. By operating with the compact
representation of the data, it is theoretically possible to produce
approximate solutions to certain large problems very efficiently.

Recently, it has been observed that dimension reduction has powerful
applications in numerical linear algebra and numerical analysis. This talk
provides a high-level introduction to randomized methods for computing
standard matrix approximations, and it summarizes a new analysis that offers
(nearly) optimal bounds on the performance of these methods. In practice,
the techniques are so effective that they compete withor even
outperformclassical algorithms. Since matrix approximations play a
ubiquitous role in areas ranging from information processing to scientific
computing, it seems certain that randomized algorithms will eventually
supplant the standard methods in some application domains.

Joint work with Gunnar Martinsson and Nathan Halko. The paper is available
at